String field theory Hamiltonians from Yang-Mills theories

Marchesini showed that the Fokker–Planck Hamiltonian for Yang-Mills theories is the loop operator. Jevicki and Rodrigues showed that the Fokker– Planck Hamiltonian of some matrix models cöıncides with temporal gauge non-critical string field theory Hamiltonians constructed by Ishibashi and Kawai (and their collaborators). Thus the loop operator for Yang–Mills theory is the temporal gauge Hamiltonian for a noncritical string field theory, in accord with Polyakov’s conjecture. The consistency condition of the string interpretation is the zigzag symmetry emphasized by Polyakov. Several aspects of the noncritical string theory are considered, relating the string field theory Hamiltonian to the worldsheet description. Mandelstam [1] realized the importance of gauge invariant loop observables in Yang–Mills theory. See also [2]. The fact that a string interpretation for Yang–Mills theories is natural is particularly transparent in the loop equation derived by Guerra and collaborators [3], and rediscovered in [4]. This Schwinger–Dyson equation governs the dynamics of gauge invariant Wilson loops in Yang–Mills theory. There are simple geometric interpretations for the various terms that appear in terms of string propagation and interactions, with the string joining interaction suppressed by a factor of 1/N relative to the string splitting interaction in a normalization natural for the large N limit. The loop equation governing Wilson loop expectation values in gauge theories can be written as the expectation value of the action of an operator, the loop operator, acting on Wilson lines. It is a fundamental observation due to Marchesini [5] that the loop operator cöıncides with the Fokker–Planck Hamiltonian that appears in the stochastic quantization [6] of the gauge theory. As such, the loop operator plays a central rôle in gauge theories but its significance in, for example, a string field theory equivalent to a Yang–Mills theory has not been elucidated. The contribution of this paper is to point out that the loop operator is precisely the temporal gauge string field theory Hamiltonian. I show that the consistency condition for this identification is Polyakov’s zig–zag symmetry. I explain how the peculiar asymmetry between joining and splitting vertices can be accounted for in a worldsheet description, and I explain how recent conjectures on Yang–Mills theories and their dimensional reductions [7–9] are related to the explicit identification given here. The explicit connection given in this paper between string theories and gauge theories should be compared to the efforts that have gone into attempted constructions of worldsheet descriptions of first–quantized string propagation in supergravity backgrounds [12] believed to be dual to gauge theories. The conceptual simplicity of the string field theory Hamiltonian